Theory of The Generalized Bernoulli-Hurwitz Numbers for The Algebraic Functions of Cyclotomic Type and The Universal Bernoulli Numbers

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Hurwitz numbers are the Laurent coefficients of an elliptic function (u) of cyclotomic type, and they are natural generalization of the Bernoulli numbers. This paper gives new generalization of Bernoulli and Hurwitz numbers for higher genus cases. They satisfy completely von Staudt-Clausen type theorem, an extension of von Staudt second theorem, and Kummer type congruence relation. The present paper is revised and combined version of math.NT/0304377 and math.NT/0312178 containing many numerical examples.